Theorem f1ocnv2d 6886

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
f1od.1	|- F = ( x e. A |-> C )
f1o2d.2	|- ( ( ph /\ x e. A ) -> C e. B )
f1o2d.3	|- ( ( ph /\ y e. B ) -> D e. A )
f1o2d.4	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )

Assertion

Ref	Expression
f1ocnv2d	|- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

Distinct variable groups:   x, y, A   x, B, y   y, C   x, D   ph, x, y

Allowed substitution hints:   C( x)   D( y)   F( x, y)

Proof of Theorem f1ocnv2d

Step	Hyp	Ref	Expression
1		f1od.1	. 2 |- F = ( x e. A |-> C )
2		f1o2d.2	. 2 |- ( ( ph /\ x e. A ) -> C e. B )
3		f1o2d.3	. 2 |- ( ( ph /\ y e. B ) -> D e. A )
4		eleq1a 2696	. . . . . 6 |- ( C e. B -> ( y = C -> y e. B ) )
5	2, 4	syl 17	. . . . 5 |- ( ( ph /\ x e. A ) -> ( y = C -> y e. B ) )
6	5	impr 649	. . . 4 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> y e. B )
7		f1o2d.4	. . . . . . . 8 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )
8	7	biimpar 502	. . . . . . 7 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ y = C ) -> x = D )
9	8	exp42 639	. . . . . 6 |- ( ph -> ( x e. A -> ( y e. B -> ( y = C -> x = D ) ) ) )
10	9	com34 91	. . . . 5 |- ( ph -> ( x e. A -> ( y = C -> ( y e. B -> x = D ) ) ) )
11	10	imp32 449	. . . 4 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B -> x = D ) )
12	6, 11	jcai 559	. . 3 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B /\ x = D ) )
13		eleq1a 2696	. . . . . 6 |- ( D e. A -> ( x = D -> x e. A ) )
14	3, 13	syl 17	. . . . 5 |- ( ( ph /\ y e. B ) -> ( x = D -> x e. A ) )
15	14	impr 649	. . . 4 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> x e. A )
16	7	biimpa 501	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ x = D ) -> y = C )
17	16	exp42 639	. . . . . . 7 |- ( ph -> ( x e. A -> ( y e. B -> ( x = D -> y = C ) ) ) )
18	17	com23 86	. . . . . 6 |- ( ph -> ( y e. B -> ( x e. A -> ( x = D -> y = C ) ) ) )
19	18	com34 91	. . . . 5 |- ( ph -> ( y e. B -> ( x = D -> ( x e. A -> y = C ) ) ) )
20	19	imp32 449	. . . 4 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A -> y = C ) )
21	15, 20	jcai 559	. . 3 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A /\ y = C ) )
22	12, 21	impbida 877	. 2 |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )
23	1, 2, 3, 22	f1ocnvd 6884	1 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   `'ccnv 5113   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  f1o2d  6887  negf1o  10460  negiso  11003  iccf1o  12316  bitsf1ocnv  15166  grpinvcnv  17483  grplactcnv  17518  issrngd  18861  opncldf1  20888  txhmeo  21606  ptuncnv  21610  icopnfcnv  22741  iccpnfcnv  22743  xrge0iifcnv  29979  rfovcnvf1od  38298